The present invention relates to motions of marine vessels while interacting with environmental waves, more particularly to methodologies for modeling such motions.
A seagoing ship is characterized by motion describable in terms of six degrees of freedom, viz., heave, surge, sway, roll, pitch, and yaw. More generally, the term “six degrees of freedom” is conventionally used to describe both translational motion and rotational motion of a body with respect to three perpendicular axes in three-dimensional space. Regarding motion of ships, the three kinds of translational ship motion are commonly referred to as heave (linear movement along a vertical axis), surge (linear movement along a horizontal fore-and-aft axis), and sway (linear movement along a horizontal port-and-starboard axis); the three kinds of rotational ship motion are commonly referred to as roll (rotational movement about a horizontal fore-and-aft axis), pitch (rotational movement about a horizontal port-and-starboard axis), and yaw (rotational movement about a vertical axis).
Ship motion at sea has always been among the most important problems for naval architects and engineers. Accurate prediction of the ship motions in real time can be critical for preventing large amplitude ship motions or the capsizing of vessels. This is particularly so for some modern day marine vessels such as fast passenger ferries, high-powered naval vessels, and high-powered cargo ships.
The history of ship motion research can be traced back more than a half century. In 1953, St. Denis and Pierson first proposed a statistical ship response in a realistic seaway by approximating ship motion in a linear system. Their work provided a respected tool to naval architects for estimating ship motions. Peters and Stoker (1957) then developed the first analytical prediction theory; see A. S. Peters and J. J. Stoker, “The Motion of a Ship, As a Floating Rigid Body, in a Seaway,” Commun. Pure Appl. Math., 10: 399-490, 1957. Because in their theory the ship's beam and unsteady motions are assumed comparable and small, Peters and Stokers' theory is called a first-order theory. Later Newman (1961) improved the first-order theory by introducing a set of small parameters and better body boundary conditions; see J. N. Newman, “A Linearized Theory for the Motion of a Thin Ship in Regular Waves,” J. Ship Res. 3(1): 1-9, 1961.
Ogilvie and Tuck (1969) developed a “strip theory” in which the linear ship motion coefficients are introduced based on the slender-body assumption, such as added masses and damping coefficients for heave and pitch motions; see T. F. Ogilvie and O. Tuck, “A Rational Strip Theory of Ship Motions: Part I,” Report-13, Dept. of the Navy, Arch. Mar. Eng., University of Michigan, Ann Arbor, pages 92+, 1969. Since Ogilvie and Tuck's strip theory includes terms containing some surface integration over the free surface, it is very difficult to implement computationally. There is an inconsistency in the strip theory: the formulation is applicable in the short wavelength domains, while the slender-body approximation works in long wavelength domains. Improvements in strip theory were developed to reconcile the difference existing within Tuck's strip theory. Notably, Maruo (1970) developed an interpolation theory, and later Newman (1978) developed a unified theory, so that the first-order theory approach can be applied over wider frequency domains; see H. Maruo, “An Improvement of the Slender Body Theory for Oscillating Ships with Zero Forward Speed,” Bull. Fac. Eng. Yokohama Nat'l Univer. 19: 45-56, 1970; J. N. Newman, “The Theory of Ship Motions,” Advances in Applied Mechanics 18: 221-83, 1978.
As faster and more powerful computers have become available, the Neumann-Kelvin approach (which basically involves strip theory) has been adopted in numerical modeling, such as disclosed by Beck and Magee (1990), Magee (1994), and Shin et al. (1997); see R. F. Beck, R. F. and A. R. Magee, “Time-Domain Analysis for Predicting Ship Motions,” Dynamics of Marine Vehicles and Structure in Waves, W. G. Price, P. Temarel & A. J. Keane, Eds., Elvesier Sciences Publishers B. V., pages 49-64, 1990; A. Magee, “Seakeeping Applications using a Time-Domain Method,” Proc. 20th Symp. Naval Hydro., Santa Barbara, Calif., 19 pages, 1994; Y.-S. Shin, J. S. Chung, W. M. Lin, S. Zhang and A. Engle, “Dynamic Loadings for Structural Analysis of Fine Form Container Ship Based on a Non-Linear Large Amplitude Motions and Loads Method,” Trans. SNAME 105: 127-54, 1997. Although Neumann-Kelvin is a linear theory, it has been a great success in ship motion research because it can be applied to arbitrary exact ship surfaces, and because the hull boundary conditions can be satisfied on the exact wetted surface of the ship body with the linearized free-surface boundary conditions.
Nevertheless, the Neumann-Kelvin theory has its own problems, such as in solving a forward-speed Green function in finite water depth; see R. F. Beck and A. M. Reed, “Modern Seakeeping Computations for Ships,” Proceedings for 23rd Naval Hydrodynamics Conference, Val-de-Reuil, France, National Academy of Sciences, pages 1-43, 2000. Still, as pointed out by Beck and Reed, probably about eighty percent of design related to ship motion calculation is based on the strip theory because, compared to other theories, it is fast, reliable, and able to accommodate various hull forms. In fact, all linear theories are ultimately beset with limitations; in particular, they cannot model any nonlinear processes. In many cases, the nonlinear processes dominate the linear effects in determining ship motions. To address this limitation, several “blending methods” were developed. For a detailed discussion, the reader is referred to the ISSC report on “Extreme Hull Girder Loading,” Committee VI.1 Report, 14th International Ship and Offshore Structures Congress 2000, Nagasaki, Japan, 59 pages, 2000.
More recently, Wilson et al. (1998) and Gentaz et al. (1999) attempted to solve the Reynolds Averaged Navier-Stokes equations in time domain (RANS); see R. Wilson, E. Paterson and F. Stern, “Unsteady RANS CFD Method for Naval Combatants in Waves,” Proc. Symp. Naval Hydro., 22nd, Washington, D.C., Washington, D.C.: Natl. Acad. Press, pages 532-49, 1998; L. Gentaz, P. E. Guillermo, B. Alessandrini and G. Delhommeau, “Three-dimensional Free-Surface Viscous Flow around a Ship in Forced Motion,” Proc. 7th International Conf. Num. Ship Hydro., Paris, France, 12 pages. Wilson et al. and Gentaz et al. applied iterative methods for steady solutions, and time-stepping methods for unsteady solutions. Their approaches are by far inconclusive, partly due to insufficient numerical results. In general, Wilson et al. and Gentaz et al. have convergence problems when the sea environment is rough and the ship forward speed is high.
According to traditional ship motion analysis, the translational and rotational solid-body motions (six degrees of freedom) of ships are approximated with parameterized forces. The traditional formulation is straightforward but is dynamically inconsistent because the forces involved are very complicated and correspond to each other in accordance with the dynamics of the ship motion. These complex forces are associated with interactions between the ship and surface waves and with departures of the ship from its equilibrium position. In principle, appropriate parameterization of such complicated dynamical forces is impossible. Consequently, it is common in conventional practice to introduce additional parameters lacking solid physics foundations in order to emulate real ship solid-body motions. An unwanted side effect of this approach is that the introductions of additional but unnecessary variables hamper computational efficiencies.
Many studies have been made on the nonlinearity of ship motion. For example, Y. Liu et al., “A High-Order Spectral Method for Nonlinear Wave-Body Interactions,” J. Fluid Mech. 245:115-136 (1992), uses a high-order spectral method to study nonlinear interactions between the ship and the water surface. See also, W. M. Lin et al., “Numerical Solution for Large-Amplitude Ship Motions in the Time-Domain,” Proc. 18th Symp. Naval Hydrod., U. Michigan, Ann Arbor, Mich. (1990); W. M. Lin et al., “Large-Amplitude Motions and Wave Loads for Ship Design,” Proc. 20th Symposium on Naval Hydrodynamics, Santa Barbara, Calif. (1994); Y. Liu et al., “Computations of Fully Nonlinear Three-Dimensional Wave-Wave Wave-Body Interactions, Part 2. Nonlinear Waves And Forces,” J Fluid Mech, 438:41-66 (2000); M. H. Xue et al., “Computations of Full Nonlinear Three-Dimensional Wave-Wave and Wave-Body Interactions, Part I. Dynamics of Steep Three-Dimensional Waves,” J. Fluid Mech, 438:11-39 (2001).
Unfortunately, there are still empirical or linear parameters used in modeling six-degree-freedom ship motions, e.g., added mass and damping mass for each degree of motion. The intrinsic limitations of these parameters can potentially affect the accuracy of the nonlinear ship motion models. For example, the added mass and the damping mass are often determined from the ship's geometry below the waterline in calm water; see A. R. Magee and R. F. Beck, Compendium of Ship Motion Calculations Using Linear Time-Domain Analysis, Department of Naval Architecture and Marine Engineering, College of Engineering University of Michigan (1988), e.g., page 192. However, the ship's geometry below the waterline when the ship is underway in waves is very different from that in calm water, and is continually changing due to speed and wave action.
It is therefore desirable to develop a new ship motion model: that maintains similar advantages of the strip theory but also efficiently models all nonlinear processes in ship motions; that is based on the fundamental laws of physics and precise mathematical formulations; that is consistent with variations of water environments and ship properties; and, that is capable of simulating ship position variations due to ship-wave interactions.